Given the sequence converges to $p^*$, show that it converges linearly: $p_{n+1}=\frac{1}{2}ln(p_n+1)$, $p_0=1$, and the limit is $p^*=0$.
I want to use fixed point theorem and denote $p_{n+1}=g(p_n)$ to show that $g'(p^*)\neq0$. But I'm not sure if it is the right approach to do so.
Thanks for any help!
It is easy to check the conditions of Banach's fixed point theorem, for example on the interval $[0,1]$, which proves convergence of the given sequence. On the other hand, since $g'(x)\ne 0$ in $[0,1]$, $g'(p^*)\ne 0$ and the result follows. In summary, you have a good plan!